Advertisements
Advertisements
प्रश्न
A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.
Advertisements
उत्तर
Let the side of the square to be cut off be x cm.
Then, the length and the breadth of the box will be (18 − 2x) cm each and height of the box will be x cm.
Volume of the box, V(x) = x(18 − 2x)2
\[V'\left( x \right) = \left( 18 - 2x \right)^2 - 4x\left( 18 - 2x \right)\]
\[ = \left( 18 - 2x \right)\left( 18 - 2x - 4x \right)\]
\[ = \left( 18 - 2x \right)\left( 18 - 6x \right)\]
\[ = 12\left( 9 - x \right)\left( 3 - x \right)\]
\[V''\left( x \right) = 12\left( - \left( 9 - x \right) - \left( 3 - x \right) \right)\]
\[ = - 12\left( 9 - x + 3 - x \right)\]
\[ = - 24\left( 6 - x \right)\]
\[\text { For maximum and minimum values of V, we must have }\]
\[ V'\left( x \right) = 0\]
\[\Rightarrow\] x = 9 or x = 3
If x = 9, then length and breadth will become 0.
∴ x ≠ 9
\[\Rightarrow\] x = 3
Now,
\[V''\left( 3 \right) = - 24\left( 6 - 3 \right) = - 72 < 0\]
∴ x = 3 is the point of maxima.
\[V\left( x \right) = 3 \left( 18 - 6 \right)^2 = 3 \times 144 = 432 {cm}^3\]
Hence, if we remove a square of side 3 cm from each corner of the square tin and make a box from the remaining sheet, then the volume of the box so obtained would be the largest, i.e. 432 cm3
APPEARS IN
संबंधित प्रश्न
f(x) = | sin 4x+3 | on R ?
f(x)=2x3 +5 on R .
f(x) = x3 \[-\] 6x2 + 9x + 15 .
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
`f(x) = 2/x - 2/x^2, x>0`
f(x) = xex.
`f(x) = x/2+2/x, x>0 `.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Find the absolute maximum and minimum values of a function f given by `f(x) = 12 x^(4/3) - 6 x^(1/3) , x in [ - 1, 1]` .
Find the absolute maximum and minimum values of a function f given by f(x) = 2x3 − 15x2 + 36x + 1 on the interval [1, 5].
How should we choose two numbers, each greater than or equal to `-2, `whose sum______________ so that the sum of the first and the cube of the second is minimum?
A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .
Find the point at which M is maximum in a given case.
Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\]. What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?
A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.
An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
Manufacturer can sell x items at a price of rupees \[\left( 5 - \frac{x}{100} \right)\] each. The cost price is Rs \[\left( \frac{x}{5} + 500 \right) .\] Find the number of items he should sell to earn maximum profit.
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and x > 0 .
Write the maximum value of f(x) = x1/x.
The maximum value of x1/x, x > 0 is __________ .
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
The function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
The minimum value of x loge x is equal to ____________ .
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
